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Differentiation and marginal analysis sheet
Essay frame

A firm's cost is C(x) = 2,000 + 40x + 0.05x² and its revenue R(x) = 150x − 0.2x². (a) Find MC and MR. (b) Evaluate them at x = 200 and interpret. (c) Find the profit at x = 200 and the marginal profit.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Differentiate the cost. MC = 40 + 0.1x.
  2. 2 · Differentiate the revenue. MR = 150 − 0.4x.
  3. 3 · Evaluate at 200. MC = 60, MR = 70.
  4. 4 · Interpret. The 201st unit costs about Rs. 60 and brings in about Rs. 70.
  5. 5 · Profit at 200. R(200) = 30,000 − 8,000 = 22,000; C(200) = 2,000 + 8,000 + 2,000 = 12,000; P = Rs. 10,000.
  6. 6 · Marginal profit. MR − MC = 10 > 0, so profit still rises with output.
  7. 7 · Business link. A manager keeps expanding while MR > MC; the next topic finds where they meet.